Chapter 9 -- Measurement of Small Current Signals--Measurement System Model and Physical Limitations
9 - 3
Unfortunately technology limits high impedance measurements because:
• Current measurement circuits always have non-zero input capacitance, i.e. C
in
> 0.
• Infinite R
in
cannot be achieved with real circuits and materials.
• Amplifiers used in the meter have input currents, i.e. I
in
> 0.
• The cell and the potentiostat create both a non-zero C
shunt
and a finite R
shunt
.
Additionally, basic physics limits high impedance measurements via Johnson noise, which is the inherent noise
in a resistance.
Johnson Noise in Z
cell
Johnson noise across a resistor represents a fundamental physical limitation. Resistors, regardless of
composition, demonstrate a minimum noise for both current and voltage, per the following equations:
E = (4 k T R δF)
1/2
I = (4 k T δF / R)
1/2
Where;
k = Boltzman's constant 1.38x 10
-23
J/
o
K
T = temperature in
o
K
δF = noise bandwidth in Hz
R = resistance in ohms.
For purposes of approximation, the Noise bandwidth, δF, is equal to the measurement frequency. Assume a
10
11
ohm resistor as Z
cell
. At 300
o
K and a measurement frequency of 1 Hz this gives a voltage noise of 41 µV
rms. The peak-to-peak noise is about 5 times the rms noise. Under these conditions, you can make a voltage
measurement of ± 10 mV across Z
cell
with an error of about ± 0.4%. Fortunately, an AC measurement can
reduce the bandwidth by integrating the measured value at the expense of additional measurement time. With
a noise bandwidth of 1 mHz, the voltage noise falls to about 1.3 µV rms.
Current noise on the same resistor under the same conditions is 0.41 fA. To place this number in perspective, a
± 10 mV signal across this same resistor will generate a current of ± 100 fA, or again an error of up to ± 0.4%.
Again, reducing the bandwidth helps. At a noise bandwidth of 1 mHz, the current noise falls to 0.013 fA.
With E
s
at 10 mV, an EIS system that measures 10
11
ohms at 1 Hz is about 2 ½ decades away from the Johnson
noise limits. At 10 Hz, the system is close enough to the Johnson noise limits to make accurate measurements
impossible. Between these limits, readings get progressively less accurate as the frequency increases.
In practice, EIS measurements usually cannot be made at high enough frequencies that Johnson noise is the
dominant noise source. If Johnson noise is a problem, averaging reduces the noise bandwidth, thereby
reducing the noise at a cost of lengthening the experiment.
Finite Input Capacitance
C
in
in Figure 9-1 represents unavoidable capacitances that always arise in real circuits. C
in
shunts R
m
, draining
off higher frequency signals, limiting the bandwidth that can be achieved for a given value of R
m
. This
calculation shows at which frequencies the effect becomes significant. The frequency limit of a current
measurement (defined by the frequency where the phase error hits 45
o
) can be calculated from:
f
RC
= 1/ ( 2 π R
m
C
in
)
Decreasing R
m
increases this frequency. However, large R
m
values are desirable to minimize the effects of
voltage drift and voltage noise in the I/E converter’s amplifiers.
A reasonable value for C
in
in a practical, computer controllable, low current measurement circuit is 20 pF. For
a 6 nA full scale current range, a practical estimate for R
m
is 10
7
ohms.
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